Proc. Natl. Acad. Sci. USAVol. 94, pp. 4229–4232, April 1997Economic Sciences

The capital-asset-pricing model and arbitrage pricingtheory: A unification

M. ALI KHAN* AND YENENG SUN†‡

*Department of Economics, Johns Hopkins University, Baltimore, MD 21218; †Department of Mathematics, National University of Singapore, Singapore 119260;and ‡Cowles Foundation, Yale University, New Haven, CT 06520

Communicated by Paul A. Samuelson, Massachusetts Institute of Technology, Cambridge, MA, October 3, 1996 (received for review August 14, 1996)

ABSTRACT We present a model of a financial market inwhich naive diversification, based simply on portfolio size andobtained as a consequence of the law of large numbers, isdistinguished from efficient diversification, based on mean-variance analysis. This distinction yields a valuation formulainvolving only the essential risk embodied in an asset’s return,where the overall risk can be decomposed into a systematic andan unsystematic part, as in the arbitrage pricing theory; andthe systematic component further decomposed into an essen-tial and an inessential part, as in the capital-asset-pricingmodel. The two theories are thus unified, and their individualasset-pricing formulas shown to be equivalent to the pervasiveeconomic principle of no arbitrage. The factors in the modelare endogenously chosen by a procedure analogous to theKarhunen–Loeve expansion of continuous time stochasticprocesses; it has an optimality property justifying the use ofa relatively small number of them to describe the underlyingcorrelational structures. Our idealized limit model is based ona continuum of assets indexed by a hyperfinite Loeb measurespace, and it is asymptotically implementable in a setting witha large but finite number of assets. Because the difficulties inthe formulation of the law of large numbers with a standardcontinuum of random variables are well known, the modeluncovers some basic phenomena not amenable to classicalmethods, and whose approximate counterparts are not al-ready, or even readily, apparent in the asymptotic setting.

Modern asset pricing theories rest on the notion that theexpected return of a particular asset depends only on thatcomponent of the total risk embodied in it that cannot bediversified away [refs. 1 and 2 (pp. 173–197)]. A marketequilibrium, by definition, precludes a price system underwhich diversification earns a reward, and thus, in a world ofcostless arbitrage, the fundamental question for asset pricingreduces to the identification and measurement of the relevantcomponent of risk that exercises influence on an asset’sexpected return.In the capital-asset-pricing model (CAPM; as in refs. 3 and

4), a particular mean-variance efficient portfolio is singled outand used as a formalization of essential risk in the market as awhole, and the expected return of an asset is related to itsnormalized covariance with this market portfolio—the so-called beta of the asset. The residual component in the totalrisk of a particular asset, inessential risk, does not earn anyreward because it can be eliminated by another portfolio withan identical cost and return but with lower level of risk (3–8).On the other hand, in the arbitrage pricing theory (APT; as inref. 9), a given finite number of factors is used as a formal-ization of systematic risks in the market as a whole, and the

expected return of an asset is related to its exposure to eachof these factors, and now summarized by a vector of factorloadings. The reward to the residual component in the returnto a particular asset, unsystematic or idiosyncratic risk, can bemade arbitrarily small simply by considering portfolios with anarbitrarily large number of assets.The basic point, however, is that the two theories capture

two different sets of risks and address different aspects of thepremium-awarding scheme for taking such risks. The CAPM,by its emphasis on efficient diversification in the context of afinite number of assets, neglects unsystematic risks in the senseof the APT; whereas the APT, with its explicit focus onmarkets with a ‘‘large’’ number of assets, and by its emphasison naive diversification and on the law of large numbers,neglects essential risks. The two theories seem to be inherentlydisjoint. It is surprising, however, that a model which unifiestheir basic ingredients can nevertheless be found; and more-over, that it is one in which the absence of arbitrage oppor-tunities is not only sufficient, but in contrast to the literature,also necessary for the validity of the APT pricing formula. Wepresent this model here.It is easy to see why such a unification has not been

considered so far. A natural way to proceed is to work with alimit model of a financial market with a continuum of assets,to identify the ensemble of systematic risks, and within this, theessential risk emanating from a suitably constructed ‘‘market’’portfolio. Standard methods, however, do not permit anyprogress toward a limit model in which nontrivial portfolioswith genuinely unsystematic or asset-specific risks can beincluded; the difficulties associated with a version of the law oflarge numbers for a standard continuum of random variables—say the Lebesgue unit interval as an index set—are wellunderstood [see refs. 10 (theorem 2.2), and 11–13]. An alter-native way is to follow the APT literature and work with anincreasing sequence of asset markets, but here one has toovercome at least three obstacles: unsystematic risks are nevercompletely eliminated, exogenous factor structures are notsufficiently refined to yield orthogonal factor loadings, andpricing formulas are approximate with convergent require-ments on infinite series (see refs. 14–16). Under these con-siderations, it is not evident how to introduce a simple explicitformula for essential risk, or more generally, how to relateimportant portfolios to the associated factor structures.Our idealized limit model of asset pricing is based on a

hyperfinite continuum of assets (17, 18). In this setting, we canappeal to a hyperfinite analogue of the Karhunen-Loeveexpansion of continuous time stochastic processes (18–22),and derive factors endogenously from the process of assetreturns. These factors are used to formalize systematic risksand to construct a ‘‘market’’ portfolio for a further specifica-tion of essential risk. The valuation formula then shows thatthe usual claim, based on the APT, that the market onlyrewards the holding of systematic risks, is simply not sharp

The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked ‘‘advertisement’’ inaccordance with 18 U.S.C. §1734 solely to indicate this fact.

Copyright q 1997 by THE NATIONAL ACADEMY OF SCIENCES OF THE USA0027-8424y97y944229-4$2.00y0PNAS is available online at http:yywww.pnas.org.

Abbreviations: CAPM, capital-asset-pricing model; APT, arbitragepricing theory.

4229

enough; systematic risk can be reduced still further until aportfolio has only essential risk, and it is only this componentof systematic risk that earns a premium. Even in the presenceof many sources of industry-wide or market-wide factor risks,there is surprisingly only a unique source of risk, characterizedby one random variable, that is rewarded by the market, andthe risk premium assigned to a particular factor risk onlyreflects the role of that factor in the definition of essential risk.In brief, our study clarifies three different types of risks infinancial markets, while previous work only focussed on two ofthem; each, one at a time, and in a different setting. Our resultsare asymptotically implementable; asymptotic arguments, be-sides being pervasive in actuarial situations (ref. 23, especiallythe final two paragraphs), furnish a valuable check on mea-surability and other assumptions that are imposed on the limit.Details of proofs will be presented elsewhere.The work reported here does not simply provide a frame-

work in which epsilons can be rigorously equated to zero, orsomewhat less naively, rely on a particular stochastic processfor which the law of large numbers holds, as constructed forexample in refs. 24–27. It belongs to the genre of ideal limitmodels that illustrate phenomena obscured in the discretecase; for examples in economic science, see refs. 28 and 29 ingeneral equilibrium theory, see refs. 30 and 31 in cooperativeand noncooperative game theory, and ref. 32 in continuous-time finance.

The Underlying Framework

In ref. 18, a hyperfinite Loeb space (17) is used to modelprobabilistic phenomena involving a large number of randomvariables in situations where there is no natural topology on theindex set. Just as the set {1, 2, z z z , n}, endowed with theuniform probability measure, is a natural space of names fora situation with n assets, a hyperfinite Loeb counting measurespace provides, by the very terms of the nonstandard exten-sion, a useful framework for the situation of an increasingsequence of asset markets indexed by {{1}, {1, 2}, z z z , {1,2, z z z , n}, z z z}. Besides being asymptotically implementable,a Loeb space is a standard measure space, and therefore allowsone to invoke mathematical structures not available in thefinite, or the large but finite, case. However, the fact that it isconstituted by nonstandard entities can be altogether ignored;‡and attention focussed simply on its special properties notshared by general measure spaces.Let T be a hyperfinite set, 7 the internal power set of T, l

the internal counting probability measure on (T, 7), and (T,L(7), L(l)) the standardization of the corresponding internalprobability space, the Loeb space. We shall use T as the spaceof asset names, and another atomless Loeb space (V, L(!),L(P)) as the sample space formalizing all possible uncertainsocial or natural states relevant to the asset market. There aretwo ways of formalizing how the index set intertwines with thesample space.§ The completion of the standard product mea-sure space (T 3 V, L(7) R L(!), L(l) R L(P)) can beconstructed in the conventional way by taking the product ofthe two relevant Loeb spaces, while the Loeb product space isthe standardization (T 3 V, L(7 R !), L(l R P)) of theinternal product space (T 3 V, 7 R !, l R P). It is aninteresting fact that the former s-algebra, denoted here by 8,is strictly contained in the Loeb product s-algebra, except in

the trivial case, inapplicable here, when one of the measuresL(l) and L(P) is purely atomic [refs. 18 and 38 (example3.12.13 presents a special case of this fact)]. This observationconstitutes the technical point of departure for the approachtaken in (18) and also exploited here. If the structure ofone-period asset returns is modeled by a real-valued L(7 R!)-measurable function x, its mean by m, and the unexpectedreturn by f 5 x 2 m, the conditional expectation E(fu8)furnishes us with the key ‘‘smoothing operation’’ that we seekfor a viable formulation of the ensembles of unsystematic andsystematic risks embodied in the market as a whole. To see this,we turn to some specific results in ref. 18.We assume the process of asset returns x to have a finite

second moment, and work with the Hilbert space +2(L(l RP)) of real-valued L(l R P)-square integrable functions on T3 V. For each asset t in T, xt [ x(t, z)[ +2(L(P)), and for eachstate v in V, xv [ x(z, v) [ +2(L(l)). Consider theinfinite-dimensional analogue of the covariance matrix of assetreturns, the autocorrelation function R(t1, t2)5 *V f(t1, v)f(t2,v)dL(P), and use it as a kernel to define a compact, self-adjoint and positive semidefinite operator on +2(L(P)). Sim-ilarly, the sample autocorrelation function can be used todefine a dual operator on +2(L(l)). For each operator, let{cn}n51

` and {wn}n51` be the respective complete eigensystems,

adjusted to form an orthonormal family, and {ln2}n51` their

common, nonincreasing sequence of all the positive eigenval-ues, each eigenvalue being repeated up to its multiplicity.Theorems 1–3 in ref. 18 show that f can be expressed, for L(lR l)-almost all (t, v) [ T 3 V, as

f~t ,v! 5 x~t ,v! 2 m~t! 5 On51

`

lncn~t!wn~v! 1 e~t ,v!, [1]

where E(fu8)(t, v) 5 •n51` lncn(t)wn(v),¶ and the residual

term e has ‘‘low’’ intercorrelation and satisfies the law of largenumbers in a strong sense with E(eu8) 5 0. Since Eq. 1 can beseen as a hyperfinite analogue of the usual factor model (20,39), the wn, cn, and ln are to be called factors, factor loadings,and scaling constants.\Now, for the purpose of this paper, let us refer to a risk as

any centered random variable defined on the sample space Vand with a finite variance. As usual, we shall use its varianceto measure its level of risk. The following is a precise formal-ization of a basic intuition.

Definition 1: A centered random variable defined on thesample space is an unsystematic (idiosyncratic) risk if it hasfinite variance and is uncorrelated with xt for L(l)-almost allt [ T.

By theorem 2 in ref. 18, one can require that for each t [ T,et is uncorrelated with xs for L(l)-almost all s [ T. Thus et isan unsystematic risk for t [ T and e represents the ensembleof all unsystematic risks in the market. Since e satisfies the lawof large numbers in a strong sense, it is also easy to see that arisk is unsystematic if and only if it is uncorrelated with all thefactors wn. It is thus natural to refer to all risks perfectlycorrelated with some linear combinations of the factors, assystematic risks. Formally,

Definition 2: A centered random variable defined on thesample space exhibits systematic risk if it has finite variance‡The relevant analogy is to all those situations when the use of a

Lebesgue measure space does not depend on the Dedekind set-theoretic construction of real numbers, or on the particular construc-tion of Lebesgue measure. For details on Loeb spaces and onnonstandard analysis, refs. 17, 24–27, 33, and 34.§For the standard mathematical concepts in this and the next twoparagraphs, see ref. 35 (chapter 8), ref. 19 (chapter 8), and ref. 36(chapter 2). For the relevant version of Fubini’s theorem, see refs. 37and 38.

¶This is the analogue of the Karhunen–Loeve biorthogonal represen-tation but for a setting without a topology on the universe T of assetindices. The representation has had applications in many fields,though not in financial economics to our knowledge; see refs. 20–22and references therein.

\The infinite sum is replaced by a sum with m terms if there are onlym nontrivial factors in the market.

4230 Economic Sciences: Khan and Sun Proc. Natl. Acad. Sci. USA 94 (1997)

and is in the linear space^ spanned by all of the factors wn, n$1.

Eq. 1 now tells us that E(fu8) represents the ensemble ofsystematic risks in the market. In the context an individualasset t, its total risk ft is decomposed into two components, asystematic risk component •n51

` lncn(t)wn and an unsystem-atic risk component et.Note that the factors wn are constructed endogenously from

the given process of asset returns, and thereby respond to thecriticism of the arbitrariness of the choice of factors in the APT(40, 41). From another point of view, a principal motivationbehind factor analysis is to find a small enough set of latentvariables so that the systematic behavior of the directly ob-served random variables can be adequately identified andexplained (20, 39). If one is allowed to use only m sources ofrisk to measure the ensemble of systematic risks of the market,then the m random variables ought to be chosen in a way thatany other set of m random variables makes a smaller contri-bution towards explaining the correlational structure of theprocess of asset returns. It can be shown that this kind of bestapproximation is indeed achieved by using the firstm elementsof the set of factors {wn}.The specification of our underlying framework is now

complete, and we turn to the formulation of APT and CAPM,and to a further decomposition of systematic risks.

Results

Aportfolio p constituted from a process x is a square integrablefunction on T, its cost C(p) is given by *Tp(t)dL(l), itsexpected return by E(p) 5 *Tp(t)m(t)dL(l), and its randomreturn 5p(v) by *Tp(t)x(t, v)dL(l). The ensemble of unsys-tematic risks identified above in the market as a whole satisfya consistent version of the law of large numbers in the sensethat the sample averages of various variations of e are L(P)-almost surely equal to zero (ref. 18, theorems 1 and 3). Sucha statement can be informally expressed as ‘‘aggregationremoves individual uncertainty,’’ or in another context, ‘‘nobetting system can ever break the house’’ (18, 23). This factallows us to compute, from Eq. 1 and5p(v), the variance V(p)5 •n51

` ln2 (*Tp(t)cn(t)dL(l))2 of any portfolio p by ignoringthe unsystematic risk component. This formula shows that ariskless portfolio, namely one with zero variance, is orthogonalto all of the factor loadings cn, and that every portfolio iscompletely diversified in the sense that it embodies only factorvariance and no unsystematic variance.We can now present the basic theorem of APT, but without

the assumption of an exogenously given, exact or approximate,factor structure. It needs to be emphasized that the noarbitrage condition is not only sufficient but also necessary forthe validity of the asset pricing formula. Such a necessitycondition is surprisingly absent in the APT literature.

Definition 3: The market permits no arbitrage opportunitiesif and only if for any portfolio p, V(p) 5 C(p) 5 0 implies E(p)5 0.

THEOREM 1. The market permits no arbitrage opportunitiesN there is a sequence {tn}n50` of real numbers such that forL(l)-almost all t [ T, m(t) 5 t0 1 •n51` tncn(t).Next, we turn to our results on the equivalence between the

APT and the CAPM.

THEOREM 2. The following equivalence holds: There is aportfolioM and a real number r such that for L(l)-almost all t[T, m(t) 5 r 1 cov(xt, M) N there is a sequence {tn}n50` of realnumbers such that •n51` (tn2yln4), `, and for L(l)-almost all t[T, m(t) 5 t0 1 •n51` tncn(t).

Theorem 2 does not furnish a precise specification of theportfolio M. We can construct a portfolio I0 based on param-

eters extracted from the given market process of asset returnsx and use it to identify the essential risk embodied in therealized return of a particular asset. Let sn, mn be the innerproducts (1, cn), (m, cn), h a risk-free portfolio defined by h5(1 2 •n51

` sncn), and m0 5 (m, h)y(h, h), if h Ó 0 and m0 50 if h 5 0. We can now define the index portfolio I0 to be •n51

`

((mn 2 m0sn)yln2)cn, with a net random return X0 5 •n51` ((mn

2 m0sn)yln)wn, cost C(I0) 5 •n51` sn(mn 2 m0sn)yln2, mean

E(I0) 5 •n51` mn(mn 2 m0sn)yln2 and variance V(I0) 5 •n51

`

(mn 2 m0sn)2yln2. We can use I0 to present

Definition 4: A centered random variable defined on thesample space exhibits essential risk if it is in the linear spacegenerated by the net random return X0 of the index portfolioI0, and exhibits inessential risk if it is in the orthogonalcomplement of X0 in the space ^ of systematic risks.

We can now present a further refinement of systematic risks,and thereby a tri-partite decomposition of the total risk of anasset. If V(I0) Þ 0, let the normalized covariance of any assett with the index portfolio be given by bt 5 cov(xt, I0)yV(I0).It can be checked that Yt(v) 5 •n51

` (lncn(t) 2 ((mn 2m0sn)yln)bt)wn(v) is orthogonal to X0 and its sum withbtX0(v) is the portion of systematic risk •n51

` lncn(t)wn inasset t. Then, by Eq. 1, the total risk of asset t can be writtenas

xt~v! 2 m~t! 5 bt X0~v! 1 Yt~v! 1 et~v!. [2]

The importance of this decomposition lies in the fact that therisk premium of almost all assets is equal to the beta of theasset multiplied by the risk premium of the index portfolio.

THEOREM 3. Assume that •n51` (mn2 m0sn)2yln4 , ` and themarket is nontrivial in the sense that the expected return functionm is not the constant function m0. Then there is no arbitrage Nthere is a sequence {tn}n50` of real numbers such that forL(l)-almost all t[ T, mt5 t01 •n51` tncn(t)N for L(l)-almostall t [ T, mt 5 m0 1 cov(xt, I0) 5 m0 1 bt(E(I0) 2 m0C(I0)).

Once we move to the asymptotic setting of an increasingsequence of finite asset markets, the insights of the limit modelcan only be extracted in an approximate form. The structureof asset returns is now formalized by a triangular array ofrandom variables {xn}n51

` defined on the product space (Tn 3V, 7n R !, ln R P), Tn 5 {1, 2, z z z , n} endowed with auniform probability measure ln on its power set 7n, and (V,!, P), a fixed common probability space. For the nth market,the realized return of asset t in Tn is given by xnt, and its meanby mnt 5 *V xnt(v)dP. The square integrability restriction onportfolios in the idealized case translates to a limitation tosequences of portfolios {pn}n51

` , for which supn$1{*Tn pnt2 dln

5 (1yn) •t [ Tn pnt2 } is finite, and that on the processes of asset

returns by the requirement that there is a positive number Msuch that * *Tn3V xnt2 dL(ln R P) # M, for all n $ 1. Inparticular, in the context of each finite market in this sequence,one can draw on the raw intuition developed for the ideal case,and identify the component of unsystematic risks, the residualerror terms en(t, v), and the factors, factor loadings and scalingconstants, {wni, cni, lni} i51

sn , sn the number of factors, allpertaining to the nth market. Furthermore, sn can be chosento be much smaller than the number of assets in the precisesense that limn3` snyn 5 0, and the en’s satisfy the law of largenumbers in an approximate way. Space considerations force usto defer a fuller elaboration of the asymptotic interpretation ofthe model; we only present the asymptotic analogue of The-orem 2 for illustrative purposes.

THEOREM 4. The following equivalence holds: There is asequence {Mn}n50` of portfolios, and a sequence {rn}n51` of realnumbers such that limn3` imnt 2 (rn 1 cov(xnt, Mn))i2 5 0 Nthere is a sequence {tni} i50

sn of real numbers and a positive number

Economic Sciences: Khan and Sun Proc. Natl. Acad. Sci. USA 94 (1997) 4231

B such that for all n $ 1, •i51sn (t ni2 ylni4 ) # B and limn3` imn 2

(tn0 1 •i51sn tnicni)i2 5 0.

Concluding Remarks

The concrete nature of the idealized limit model that we reportabove allows us to explore conditions for the existence ofvarious important portfolios—risk-free, factor, mean, cost,and mean-variance efficient—and to develop explicit formulasfor them. The hyperfinite model, besides being asymptoticallyimplementable, also exhibits a universality property in the sensethat the distributions of the individual random variables in theensemble of unsystematic risks may be allowed any variety ofdistributions (18).

It is a pleasure to acknowledge the support of Profs. Donald Brown,Lionel McKenzie, and Heracles Polemarchakis and the interest andsuggestions of two referees. This work was presented at the Universityof Notre Dame (April 22, 1996), Universidade Nova de Lisboa(November 22, 1996), University of Rochester (December 11, 1996),Rutgers University (March 3, 1997) and Columbia University (March7, 1997); the comments and questions of Ralph Chami, Tom Cosi-mano, Jaksa Cvitanic, Ioannis Karatzas, Rich McLean, Paulo Mon-teiro, Vasco d’Orey, Mario Pascoa, and Kali Rath are gratefullyacknowledged. Y.S. is also grateful to the mathematical financeinterest group at the National University of Singapore for helpfulconversations and stimulating seminars. Part of the research was donewhen Y.S. visited the Department of Economics at Johns Hopkinsduring June–July 1995; we both thank Lou Maccini and the Depart-ment for support.

1. Ross, S. A., Westerfield, R. W. & Jaffe, J. (1996) CorporateFinance (Richard Irwin, Chicago).

2. Radcliffe, R. C. (1994) Investment (Harper Collins, New York).3. Sharpe, W. (1964) J. Finance 33, 885–901.4. Lintner, J. (1965) Rev. Econ. Stat. 47, 13–37.5. Markowitz, H. M. (1959) Portfolio Selection, Efficient Diversifi-

cation of Investments (Wiley, New York).6. Tobin, J. (1958) Rev. Econ. Stat. 25, 65–86.7. Samuelson, P. A. (1967) J. Finance Q. Anal. 2, 1–13; 107–122.8. Samuelson, P. A. (1970) Rev. Econ. Stud. 37, 537–542.9. Ross, S. A. (1976) J. Econ. Theory 13, 341–360.10. Doob, J. L. (1937) Trans. Am. Math. Soc. 37, 107–140.11. Doob, J. L. (1953) Stochastic Processes (Wiley, New York).12. Judd, K. L. (1985) J. Econ. Theory 35, 19–25.13. Feldman, M. & Gilles, C. (1985) J. Econ. Theory 35, 26–32.14. Huberman, G. (1982) J. Econ. Theory 28, 183–191.

15. Huberman, G. (1987) in The New Palgrave, eds. Eatwell, J.,Newman, P. K. & Milgate, M. (Macmillan, London).

16. Chamberlain, G. & Rothschild, M. (1983) Econometrica 51,1281–1304.

17. Loeb, P. A. (1975) Trans. Am. Math. Soc. 211, 113–122.18. Sun, Y. N. (1996) Bull. Sym. Logic 2, 189–198.19. Loeve, M. (1977) Probability Theory (Springer, New York), 4th

Ed., Vols. I and II.20. Basilevsky, A. (1994) Statistical Factor Analysis and Related

Methods (Wiley, New York).21. Oja, E. (1983) Subspace Methods of Pattern Recognition (Wiley,

New York).22. Fukunaga, K. & Koontz, W. L. G. (1970) Infor. Control 16,

85–101.23. Samuelson, P. A. (1963) Scientia 57, 1–6.24. Keisler, H. J. (1984) Mem. Am. Math. Soc. 48, 1–184.25. Anderson, R. M. (1991) in Handbook of Mathematical Econom-

ics, eds. Hildenbrand, W. & Sonnenschein, H. (North Holland,Amsterdam), pp. 2145–2218.

26. Nelson, E. (1987) Radically Elementary Probability Theory(Princeton Univ. Press, Princeton).

27. Stroyan, K. D. & Bayod, J. M. (1986) Foundations of InfinitesimalStochastic Analysis (North Holland, New York).

28. Aumann, R. J. (1964) Econometrica 34, 1–17.29. Brown, D. J. & Robinson, A. (1972) Proc. Natl. Acad. Sci. USA

69, 1258–1260, and correction (1972) 69, 3068.30. Milnor, J. W. & Shapley, L. S. (1961) Reprinted (1978) Math.

Oper. Res. 3, 290–307.31. Schmeidler, D. (1973) J. Stat. Phys. 7, 295–300.32. Merton, R. C. (1990) Continuous-Time Finance (Basil Blackwell,

Oxford).33. Cutland, N. (1983) Bull. London Math. Soc. 15, 529–589.34. Lindstrom, T. L. (1988) in Nonstandard Analysis and its Applica-

tions, ed. Cutland, N. (Cambridge Univ. Press, Cambridge),1–106.

35. Rudin,W. (1974)Real and Complex Analysis (McGraw–Hill, NewYork).

36. Kadison, R. V. & Ringrose, J. R. (1983) Fundamentals of theTheory of Operator Algebras (Academic, New York).

37. Keisler, H. J. (1988) Nonstandard Analysis and its Applications,ed. Cutland, N. (Cambridge University Press, Cambridge), pp.106–139.

38. Albeverio, S., Fenstad, J. E., Hough-Krohm, R. & Lindstrom,T. L. (1986) Nonstandard Methods in Stochastic Analysis andMathematical Physics (Academic, Orlando, FL).

39. Harman, H. H. (1968)Modern Factor Analysis (Univ. of ChicagoPress, Chicago), 2nd Ed.

40. Gilles, C. & LeRoy S. F. (1991) Econ. Theory 1, 213–230.41. Brown, S. J. (1989) J. Finance 44, 1247–1262.

4232 Economic Sciences: Khan and Sun Proc. Natl. Acad. Sci. USA 94 (1997)